This conjecture is made based on a number of examples in the discrete setting. Let \(A, B, C\)  be finite sets in \(\mathbb {R}\) . It is well-known that one can use the Szemerédi-Trotter theorem [1]} to show that if \(|B||C|\gtrsim |A|\)  then there exists \(c\in C\)  such that \(|A+cB|\gtrsim |A|\) . For reader's convenience, we reproduce the argument here. For any \(c\in C\) , let \(L_c\)  be the set of lines of the form \(x=r-cy\)  with \(r\in A+cB\) . Let \(L=\cup _{c\in C}L_c\) . Then it is clear that \(|L|\le |C|\max _{c\in C}|A+cB|\) . We observe that \(I(A\times B, L)\ge |A||B||C|\) . Thus, the Szemerédi-Trotter incidence theorem gives
\(|A||B||C|\lesssim |A|^{2/3}|B|^{2/3}|C|^{2/3}\max _{c\in C}|A+cB|^{2/3},\) 
